Problem: Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{k^2 - 1}{k^2 + k}$
First factor the expressions in the numerator and denominator. $ \dfrac{k^2 - 1}{k^2 + k} = \dfrac{(k - 1)(k + 1)}{(k)(k + 1)} $ Notice that the term $(k + 1)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(k + 1)$ gives: $a = \dfrac{k - 1}{k}$ Since we divided by $(k + 1)$, $k \neq -1$. $a = \dfrac{k - 1}{k}; \space k \neq -1$